1.3. Operations on matrices

a_ij = b_ij    i, j

The sum of m x n matrices A = (a_ij ) and B = (b_ij ) is the m x n matrix C = (c_ij ),

whose entries are c_ij = a_ij + b_ij .

Example 1: Matrix addition

The product cA of a matrix A = (a_ij ) and a number c is the matrix (ca_ij ),

In particular - B = ( - b_ij ) is the negative of a matrix B = (b_ij ) and

The product AB (in this order) of matrices A_{m x n} = (a_ik ) and B_{n x p} = (b_kj ) is the m x p matrix C = (c_ij ), where

c_ij = a_ik b_kj .

a11 a12 ··· a1n a21 a22 ··· a2n : am1 am2 ··· amn

b11 b12 ··· b1p b21 b22 ··· b2p : : bn1 bn2 ··· bnp

=		a11 b11 + a12 b21 + ··· + a1n bn1			a11 b12 + ··· + a1n bn2  ···
		a21 b11 + a22 b21 + ··· + a2n bn1			a21 b12 + ··· + a2n bn2  ···
		:			:
		am1 b11 + am2 b21 + ··· + amn bn1			am1 b12 + ··· + amn bn2  ···

Note that the product AB is defined only when the number of the columns of A and the number of the rows of B coincide.
It is possible that AB is defined while BA is not.

Example 2: Multiplication of matrices

If AB = BA, we say that A and B commute. In this case A and B are n x n matrices and therefore they are square matrices of the same dimension.

Example 3: Commuting matrices

Proposition 1 When the operations below are defined, we have

1) A + B = B + A
2) (A + B) + C = A + (B + C)
3) A + O = A
4) A + ( - A) = O
5) (µ)A = (µ A)
6) ( + µ)A = A + µ A
7) (A + B) = A + B
8) 1· A = A
9) (AB)C = A (BC)
10) A(B + C) = AB + AC
11) (A + B)C = AC + BC
12) (AB) = (A)B = A(B)
13) IA = A
14) AI = A

Proof. Follows from the previous definitions of matrix operations.

Notice! If AB = 0, it does not necessarily follow that A = 0 or B = 0:

Example

A = 1 2 -1 0, 1 0 -1

B = 2 1 0 0 0 2 1

AB = 1 2 -1 1 0 -1

2 1 0 0 2 1

= 02x2

Also

IC = C:    1 0 0 0 1 0 0 0 1

a b c d e f g h i

= a b c = C d e f g h i

Transposition

The transpose or the transposed matrix of an m x n matrix A = (a_ij ) is the n x m matrix A^T = (a_ji ), that is, the rows of A^T are the columns of A and the columns of A^T are the rows of A.

A on symmetric if A^T = A.

A on skew symmetric if A^T = - A

Example 4: Symmetric matrices

Proposition 2

1) (A^T)^T = A
2) (A + B)^T = A^T + B^T
3) (A)^T = A^T
4) (AB)^T = B^TA^T

Problem: Show that a square matrix A can be written as the sum of a symmetric and a skew symmetric matrix: A = ½(A + A^T) + ½(A - A^T).

In applications (for example, in signal processing) one often needs real symmetric Toeplitz matrices (for example, autocorrelation matrix)

	r (0)	r (1)	r (2)	···	r (n - 1)		.
	r (1)	r (0)	r (1)	···	r (n - 2)
	r (2)	r (1)	r (0)	···	r (n - 3)
	:
	r (n - 1)	r (n - 2)	r (n - 3)	···	r (0)

If A = (a_ij ) is a complex matrix then the conjugate matrix of A is

= (_ij ).

A square matrix A is a Hermite matrix (Hermitian), if A^T = and a skew Hermite matrix (skew Hermitian) if A^T = - .

The Hermite matrix of A is A^H = ^T. If A = A^H, then A is Hermitian.

The diagonal elements of a Hermite matrix are real, because a_ii = _ii . The diagonal elements of a skew Hermite matrix are pure imaginary or zero, because a_ii = - _ii . Hermiteness generalizes the notion of symmetricness.

Example 5: Hermitian matrix

Inverse of a matrix

Matrix B is the inverse of a matrix A if

Proposition 3. If A has an inverse, the inverse is unique.

Proof. Let B and B' be inverses of a matrix A, that is, AB = BA = I, and AB' = B'A = I.
But it now follows that B = IB = (B'A)B = B' (AB) = B'I = B'.

If the inverse of A exists, it is denoted by A ^{- 1} and we say that A is regular. If A does not have an inverse, it is singular.

It can be shown that for square matrices A and B we have

Therefore, to prove that B is the inverse of A, it suffices to show that AB = I.

Example 6: An inverse matrix

Proposition 4. If A and B are regular and 0, then

1) (A ^{- 1}) ^{- 1} = A
2) (A) ^{- 1} = (1/ )A ^{- 1}
3) (AB) ^{- 1} = B ^{- 1}A ^{- 1}
4) (A^T) ^{- 1} = (A ^{- 1})^T

Proof. 4): Denote B = (A ^{- 1})^T. We have to show that A^TB = BA^T = I

A is orthogonal if it is real and A^T = A ^{- 1}. If A and B are orthogonal, then so is AB.
Proof. (AB)^T = B^TA^T = B^-1A^-1 = (AB)^-1.
An orthogonal mapping(a matrix) preserves norms!

More generally: m x n matrix U is orthogonal if U^TU = I (column orthogonal).

Example 7: Rotation and reflection matrices

A is unitary if ^T = A ^{- 1}, that is, if A ^{- 1} = A^H. Compare orthogonal vs. unitary.

Block matrices

Often it is useful to partition matrices into smaller ones, called blocks, for example

A = x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

= A11 A12 A13 A21 A22 A23

Proposition 5. Let matrices A and B be partitioned as follows:

A = A11 ··· A1n , : : Ap1 ··· Apn

B = B11 ··· B1m , : : Br1 ··· Brm

where the block A_ij is a s_i x t_j matrix and B_ij is a u_i x v_j matrix. Then

1)	AT = A11 T ··· Ap1 T : : A1n T ··· Apn T
2)	A = A11 ··· A1n : : Ap1 ··· Apn

3) If p = r, n = m and s_i = u_i , t_j = v_j    i, j, then

A + B =		C11	···	C1n
		:		:
		Cp1	···	Cpn

where C_ij = A_ij + B_ij

4) If n = r and t_j = u_j    j, then

AB =		C11	···	C1m
		:		:
		Cp1	···	Cpm

where C_ij = A_ik B_kj .

Proof. Straightforward calculation.

Example 8: Block matrices